MinT - Best Known (69−4, 69, s)-Nets in Base 25

Best Known (69−4, 69, s)-Nets in Base 25

(69−4, 69, large)-Net over F₂₅ — Constructive and digital

Digital (65, 69, large)-net over F₂₅, using

5 times m-reduction [i] based on digital (65, 74, large)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 0, 391875)-net over F₂₅, using
    - s-reduction based on digital (0, 0, s)-net over F₂₅ with arbitrarily large s, using
      - arbitrary set of 25⁰ points [i]
  2. digital (0, 0, 391875)-net over F₂₅ (see above)
  3. digital (0, 0, 391875)-net over F₂₅ (see above)
  4. digital (0, 0, 391875)-net over F₂₅ (see above)
  5. digital (0, 0, 391875)-net over F₂₅ (see above)
  6. digital (0, 0, 391875)-net over F₂₅ (see above)
  7. digital (0, 0, 391875)-net over F₂₅ (see above)
  8. digital (0, 0, 391875)-net over F₂₅ (see above)
  9. digital (0, 0, 391875)-net over F₂₅ (see above)
  10. digital (0, 0, 391875)-net over F₂₅ (see above)
  11. digital (0, 0, 391875)-net over F₂₅ (see above)
  12. digital (0, 0, 391875)-net over F₂₅ (see above)
  13. digital (0, 0, 391875)-net over F₂₅ (see above)
  14. digital (0, 0, 391875)-net over F₂₅ (see above)
  15. digital (0, 0, 391875)-net over F₂₅ (see above)
  16. digital (0, 0, 391875)-net over F₂₅ (see above)
  17. digital (0, 1, 391875)-net over F₂₅, using
    - s-reduction based on digital (0, 1, s)-net over F₂₅ with arbitrarily large s, using
      - construction for strength kÂ =Â 1 [i]
  18. digital (0, 1, 391875)-net over F₂₅ (see above)
  19. digital (0, 1, 391875)-net over F₂₅ (see above)
  20. digital (0, 1, 391875)-net over F₂₅ (see above)
  21. digital (0, 1, 391875)-net over F₂₅ (see above)
  22. digital (3, 5, 391875)-net over F₂₅, using
    - s-reduction based on digital (3, 5, 406901)-net over F₂₅, using
      - kÂ =Â 2 construction [i]
  23. digital (4, 7, 391875)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁷, 391875, F₂₅, 3, 3) (dual of [(391875, 3), 1175618, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(25⁷, 391875, F₂₅, 2, 3) (dual of [(391875, 2), 783743, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(25⁷, 391875, F₂₅, 3) (dual of [391875, 391868, 4]-code or 391875-cap in PG(6,25)), using
        product of two caps with tangent hyperplane [i]
  24. digital (12, 16, 2097150)-net over F₂₅, using
    - s-reduction based on digital (12, 16, 4194301)-net over F₂₅, using
      - net defined by OOA [i] based on linear OOA(25¹⁶, 4194301, F₂₅, 4, 4) (dual of [(4194301, 4), 16777188, 5]-NRT-code), using
        OA 2-folding and stacking [i] based on linear OA(25¹⁶, 8388602, F₂₅, 4) (dual of [8388602, 8388586, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(25¹⁶, large, F₂₅, 4) (dual of [large, large−16, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 9765624Â = 25⁵âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
  25. digital (32, 41, 2097150)-net over F₂₅, using
    - net defined by OOA [i] based on linear OOA(25⁴¹, 2097150, F₂₅, 9, 9) (dual of [(2097150, 9), 18874309, 10]-NRT-code), using
      - OOA 4-folding and stacking with additional row [i] based on linear OA(25⁴¹, 8388601, F₂₅, 9) (dual of [8388601, 8388560, 10]-code), using
        discarding factors / shortening the dual code based on linear OA(25⁴¹, large, F₂₅, 9) (dual of [large, large−41, 10]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 25¹⁰âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]

(69−4, 69, large)-Net in Base 25 — Upper bound on s

There is no (65, 69, large)-net in base 25, because

2 times m-reduction [i] would yield (65, 67, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]

Best Known (69−4, 69, s)-Nets in Base 25

(69−4, 69, large)-Net over F25 — Constructive and digital

(69−4, 69, large)-Net in Base 25 — Upper bound on s

(69−4, 69, large)-Net over F₂₅ — Constructive and digital